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Volatility Smile Volatility Smile Peter Nowak∗ and Patrick Sibetzy ∗ covering SABR Model ycovering Heston Model April 24, 2012 Abstract This document is analysing two famous stochastic volatily models, namely SABR and Heston. It introduces the problems of the Black Scholes model, the two stochastic models and in a final step calibrates volatilty smiles/- surfaces for given FX option data. Key words: smiles, skew, implied volatilities, stochastic volatilities, SABR, Heston 1 Contents 1 Introduction 3 2 Black's model with implied volatilities 4 2.1 FX Black Scholes Framework . 4 2.2 FX Implied Volatility Smile . 4 3 Stochastic volatility models 7 3.1 The Heston model . 7 3.1.1 Heston Option Price . 10 3.1.2 Characteristic Functions . 12 3.1.3 Heston FX Option Extension . 15 3.1.4 Summary for Heston Option Pricing . 17 3.2 The SABR model . 19 3.2.1 Definition . 19 3.2.2 SABR Implied Volatility . 20 3.2.3 Model dynamics . 21 3.2.4 Parameter Estimation . 22 3.2.5 SABR Refinements . 24 3.3 Calibration and Simulation . 26 3.3.1 The Heston Model . 26 3.3.2 The SABR model . 30 4 Conclusio 37 5 Appendix 39 5.1 Heston Riccati differential equation . 39 5.2 Analysis of the the SABR model . 40 5.3 Implementation in R . 46 5.3.1 FX Black Scholes Framework . 46 5.3.2 Heston Framework . 47 5.3.3 SABR Framework . 48 2 1 Introduction According to the Black-Scholes model, we should expect options that expire on the same date to have the same implied volatility regardless of the strikes. Thus, Black erroneously assumed that the volatility of the underlying is con- stant. However, implied volatilities vary among the different strike prices. This discrepancy is known as the volatility skew or smile. In general, at-the-money options tend to have lower volatilities that in- or out-of-the-money options, see figure 1. For estimating and fitting such volatility smiles, in terms to accuaratly price op- tions, several frameworks have been introduced. Merton [10] suggested to make the volatility a deterministic function of time. This would indeed explain the different volatility for different tenors, but would not explain the smile effect for different strikes. Other local volatility models introduced by Dupire [4], or the one from Derman and Kani [3], including a state dependent volatility coefficient yields still a complete market model, but it cannot explain the persistent smile shape which does not vanish over time with longer maturities. Thus the next step would be to allow the volatility to move stochastically over time, where we will chose models to fit a certain market implied volatility sur- face. This processes were pioneered by Hull and White [8], Heston [7] and later by Patrick Hagan through the widely used SABR model [6]. 3 2 Black's model with implied volatilities The implied volatility of a european option in the Black Scholes framework is an alternative of quoting the options price, as every other parameter are observable in the market. If the market price of the option is quoted one find σimp that the Black Scholes option model price CBS equals the option's market price. BS ∗ C (S; K; σimp; rd; rf ; t; T ) = C (1) As the option price is monotonically increasing in the volatility it can be uniquely determined. However, as the BSM formula cannot be solved for the volatility analytically, a numerical algorithm has to optimise this approach. For our examples we will analyse the FX option implied volatility surface, as currencies tend to provide the so called volatility smile in general. Therefore we need also to introduce the differences to the normal Black Scholes framework. 2.1 FX Black Scholes Framework For the FX smile we will consider a model for the FX spot rates to be strictly positive and evolve stochastically over time. In our model framework we will adapt the Black Scholes model [1] with the model of Garman and Kohlhagen [5] which is also based onto Black Scholes as an application to foreign currency options. The following analysis is based on risk neutral valuation, which means that a risk free portfolio will always yield the risk free interest rate in the re- spective currency. Otherwise there would be an arbitrage opportunity leading to a risk free profit. The important point of risk neutral valuation is, that the underlying asset itself is a risky investment and therefore also derivatives based on it are risky too. However, it is possible to costruct an instantaneously risk free portfolio consist- ing of the two securities. The proportions of the two securities however, are not static, but need constantly be adapted over time. This process is thus often referred to as dynamic hedging. 2.2 FX Implied Volatility Smile Perhaps due to that American options are almost not traded in FX markets, the market uses the Black Scholes formula for price quotation. These quotes are in Black Scholes implied volatilities rather than the prices directly. What is also a peculiarity of the FX market is that quotes are provided at a fixed Black 4 Scholes delta not at fixed strike levels as it is usual for options in other markets. In particular the options are quoted implicitly for five different levels of delta for different tenor points. These standard moneyness levels are at the money level, 25 delta out of the money level and 25 delta in the money level (75 delta) and the same for 10 delta. FX Volatility Smile ● RR10 ● ● BF10 ● Implied Volatility ● ATM 10C 25C ATM 25P 10P Delta Figure 1: FX Smile including the three point market convention quotation Since out of the money levels are liquid moneyness levels in the options market, market quotes these levels as 25 delta call and 25 delta put. If a trader has the right model, he can build the whole volatility smile for any time to expiry by using the three points in the volatility surface. The additional two points of 10 delta options yields a better calibration as far out of the money options can have even higher than extrapolated implied volatility. In the options market 25 delta call and 25 delta put points are not quoted as volatility. They are quoted according to their positions to at the money volatilty level. These parameters are 25 delta butterfly and 25 delta risk reversal. Risk Reversal: Risk reversal is the difference between the volatility of the call price and the put price with the same moneyness levels. 25 delta risk reversal is the difference between the volatility of 25 delta out of the money Call and 25 delta out of the money Put. 5 RR25 = σ25C − σ25P Butterfly: Butterfly is the difference between the avarage volatility of the call price and put price with the same moneyness level and at the money volatility level. In other words for example for 25 delta level, butterfly defines how far the average volatil- ity of 25 delta call and 25 delta put is away from the at the money volatiltiy level. BF25 = (σ25C + σ25P )=2 − σAT M A real world example shall motivate the necessity to apply option pricing mod- els that are richer than the classical model of Black and Scholes (1973, [1]). It shows that the Black-Scholes implied volatilities for EUR/JPY FX options for different deltas and maturities. Figure 2: Bloomberg market data for the USDJPY implied volatility surface The volatility surface of this dataset then looks the following 6 USDJPY FX Option Volatility Smile 0.15 0.14 0.13 0.12 Implied Volatility 0.11 0.10 0.09 10C 25C ATM 25P 10P Delta Figure 3: USDJPY implied volatility surface 3 Stochastic volatility models 3.1 The Heston model A well established model to price equity options including a volatility smile or skew in practice is the Heston Model. Here the underlying follows a diffusion stochastic process, like in the Black Scholes model, but the process' stochastic variance ν follows a Cox Ingersoll Ross (CIR) process. p S dSt = µStdt + νtStdWt p ν dνt = κ(θ − νt)dt + σ νtdWt S ν dWt dWt = ρdt In this model the positive volatility of the underlyings volatility σ generates a smile, and a nonzero correlation ρ generates a skew of the volatility curve with slope of the same sign. The parameters in this model are: µ the drift of the underlying process κ the speed of mean reversion for the variance θ the long term mean level for the variance 7 σ the volatility of the variance ν0 the initial variance at t = 0 ρ the correlation between the two Brownian motions To derive the semianalytic solution for the FX Option we will first begin with the classical Heston model for equity options and then show the extension to the two currency world model. We begin with a portfolio consisting of one asset more than in the Black Scholes replication approach, as we have also one more Brownian motion driving the underlying's volatility. Thus we have a portfolio consisting of one option V (S; ν; t) a portion of the underlying ∆St and a third derivative to hedge the volatility φU(S; ν; t). The portfolio has then the value Πt = V (S; ν; t) + ∆St + φU(S; ν; t): The next assumption we make is that the portfolio is selffinancing which brings us to the following equation which describes the change in value of the portfolio: dΠ = dV + ∆dS + φdU For the two Options U and V we apply the Ito-formula to expand dU(S; ν; t): 1 1 dU = U dt + U dS + U dν + U (dS)2 + U (dSdν) + U (dν)2 t S ν 2 SS Sν 2 νν With the quadratic variation and covariation terms expanded we get (dS)2 = d hSi = νS2d W S = νS2dt; (dSdν) = d hS; νi = νSσd W S;W ν = νSσρdt, and (dν)2 = d hνi = σ2νd hW ν i = σ2νdt: The other terms including d hti ; d ht; W ν i ; d t; W S are left out, as the quadratic variation of a finite variation term is always zero and thus the terms vanish.
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